Knots are not for naught: Design, properties, and topology of hierarchical intertwined microarchitected materials

Lightweight and tough engineered materials are often designed with three-dimensional hierarchy and interconnected structural members whose junctions are detrimental to their performance because they serve as stress concentrations for damage accumulation and lower mechanical resilience. We introduce a previously unexplored class of architected materials, whose components are interwoven and contain no junctions, and incorporate micro-knots as building blocks within these hierarchical networks. Tensile experiments, which show close quantitative agreements with an analytical model for overhand knots, reveal that knot topology allows a new regime of deformation capable of shape retention, leading to a ~92% increase in absorbed energy and an up to ~107% increase in failure strain compared to woven structures, along with an up to ~11% increase in specific energy density compared to topologically similar monolithic lattices. Our exploration unlocks knotting and frictional contact to create highly extensible low-density materials with tunable shape reconfiguration and energy absorption capabilities.


Tensile Responses of Structures Composed of Multiple Rhombus Frames
contains SEM images of typical knotted lattices with the inevitable printing-induced defects pointed by the red arrows. We observed similar warping in all 2x2x2 knotted structures, which is caused by the proximity of the printable build area in our two-photon polymerization system (Nanoscribe) to the edges of the 140x140x140 μm 3 lattices, as well as by the detached horizontal rhombuses. Since the activation of the knot tightening mechanism requires fibers to be able to slide past each other, the observed warping may prevent the fibers within the knots to slide smoothly and instead induce a stick-slip behavior and/or suppress fiber motion. If the fibers within a knot are unable to slide as intended, whether by sticking or constriction, the structure would essentially behave as a woven structure when pulled in tension, which is what we observed in the tensile test result of our 2x2x2 knotted structure (Fig. S1A).
To circumvent the issue of warping and fiber sticking, we designed a simpler unit cell with two knotted rhombus frames aligned vertically, termed Reduced Unit Cell (RUC), and tested single RUC and multi-RUC structures in tension as shown in Fig. 3A and Fig. S1, B and C. We varied the fiber radius * for each structure to mitigate unwanted sticking and constriction, and we estimated the resulting relative density of each structure by comparing the measured fiber radii with * = 1.69 μm for a full knotted lattice with three rhombus frames and a relative density ̅ of 5%, analogous to the one in Fig. S1A. We observed that the knot tightening mechanism was activated for multi-RUC structures (termed Reduced Lattice) in Fig. 3A, resulting in a specific energy density Ws of up to 3.9 kJ/kg, which is 11% and 70% higher than the average Ws of monolithic and woven octahedron lattices in Ref. (40), respectively.

Validity of Unknotting Number in the Model
The effect of the unknotting number in the overhand knot model manifests itself in the loop radius as a function of strain : → as → 150% for = 2 in the model, and → as → 94% for = 3. These strains represent initiation of transition from knot tightening to fiber stretching regime (Fig. 2B); beyond them the model no longer applies. We measured the highest failure strain of pristine and UV-irradiated knotted rhombuses in our experiments to be ~147%, rendering < 3 to be more appropriate for comparison with experiments.
X-ray photo-electron spectroscopy (XPS) Characterization XPS performed on as-printed and 29-hr irradiated samples revealed slight differences in peaks assigned to carbon in several oxidation states, as demonstrated in Tables S1 and S2 (see also Fig.  S4). However, the C 1s signal as assigned to the as-printed sample was not consistent between different as-printed samples as seen between Tables S1 and S3, indicating a lack of an unambiguous signal of chain scission in the system. Table 1 shows a C 1s signal composed of five elements yielded from XPS performed on a single site on one IP-Dip sample measured near the time of synthesis. Full width at half maximum (FWHM) of all peaks other than the π-π* shake-up was constrained between 0.8-1.4 eV as appropriate for a modern instrument. Notably, a stretch at the expected position of a C=O signal was present despite the absence of carbonyls in the known composition of the photoresist. This may be due to the presence of carbonyls in the undeclared 10% of the IP-dip photoresist as per the MSDS [see Refs. (65,66) for more information on conducting and interpreting XPS results]. We record the ratio of the C-O:C-C/C-H and O-C=O:C-C/C-H intensities (RSF adjusted) from the sample corresponding to Table 1  XPS performed on a single site on one IP-Dip sample irradiated for 29 hr yielded a C 1s signal composed of five elements assigned in Table 2. FWHM of all peaks other than the π-π* shake-up was constrained between 0.8-1.4 eV as appropriate for a modern instrument. Again, a stretch at the expected position of a C=O signal was present despite the absence of carbonyls in the known composition of the photoresist. With respect to the pre-irradiation sample, we noted that the ratios C-O:C-C/C-H and O-C=O:C-C/C-H with respect to peak intensities (RSF adjusted) decreased after irradiation (see below); however, given the differing results for the pre-treatment and post-treatment conditions in Tables S3 and S4, we are skeptical of the utility or clarity of this result. The ratio of the C-O:C-C/C-H and O-C=O:C-C/C-H intensities (RSF adjusted) corresponding to Table 2  The computations below, following the attempted method of quantifying the degree of transformation of C-O and O-C=O functional groups in the irradiation process, demonstrate the potentially widely varying value of the ratio of these components between scans of samples produced under identical conditions. Given the inherent difficulty in differentiating between carbon deriving from an adventitious source versus carbon in a known polymer, these values shed doubt on the ability to accurately derive transformation of surface polymer chains from the collected XPS data, especially when compared to data obtained from 29-hr UV-irradiation of the previous sample. The ratio of the C-O:C-C/C-H and O-C=O:C-C/C-H intensities (RSF adjusted) corresponding to Table S3 are  The Influence of Tensile Failure Strain of the Constituent Material on Intertwined Frames As highlighted in Fig. 2, A to C, the load-strain behavior of both the knotted and woven frames are nearly identical in the fiber alignment region (Regime 1) before diverging in subsequent regions. While knotted frames in our work generally reached higher tensile failure strains compared to their woven counterparts, passivated frames of both topologies were the only types of samples that consistently failed within Regime 1 at similar failure strains and loads. To understand the influence of the tensile failure strain of the constituent material ( UTS ) on the extensibility of intertwined frames, we first estimate the minimum UTS required to allow the structure to deform past the fiber alignment regime, i.e., the maximum tensile strain induced in a fiber when a helical strand of three fibers is straightened.
For a helix with an effective beam radius * (see Fig. 1 and Fig. S7) and a fiber radius * , the helix can be parametrically defined as ̅ = ( ℎ cos , ℎ sin , ), where ℎ = * − * is the helix radius, = 2 is the helix parameter with being the pitch of the helix, and = 2 where is the number of helix evolutions. The curvature of the helix can be calculated as and the arc length s can be calculated as Assuming the helix arc length stays constant and torsion is negligible, the maximum tensile strain , of a fiber straightened to a curvature s from an initial curvature 0 can be calculated as By taking * = 1.69 μm, the initial beam radius * 0 = 3.5 μm, and the straightened beam radius * s = 2r * √3 for the 70 μm unit cell design, we obtain a maximum tensile strain of 9.6% within a given fiber in the helix. Among the pillars that we have tested (see Fig. S2), the passivated pillars were the only samples that consistently failed in tension below the 9.6% straightening strain threshold. This result, coupled with the observation that passivated frames were the only samples that could not deform past the fiber alignment regime, suggests that the tensile failure strain of the constituent material is crucial in determining the possibility of failure of an intertwined frame within Regime 1.
For the non-passivated intertwined frames that deformed beyond the fiber alignment regime, we did not observe any correlation between their failure strains and the failure strains of their corresponding pillars despite a nearly 46% reduction in UTS between pristine pillars and 29-hr UV-irradiated ones. To better show the discrepancy between our experimental results and what we would expect if UTS were as crucial in determining frame failure as in Regime 1, we can roughly estimate the expected failure strain of the frame by calculating the change in the frame effective strain when fibers are aligned in the loading direction , the maximum tensile strain within a fiber in the junction during alignment , and the change in frame effective strain when fibers are stretched . For a rhombus frame with a 45° internal angle and height H, can be calculated as We estimate by replacing 0 in Equation R3 with the tightest curvature within the junction, which is approximately 0.17 μm -1 . While this estimate results in = 27.2%, which is around 14% higher than the tensile failure strain of 29-hr UV-irradiated pillars, it also assumes that the fibers in the strands are all straightened to tightly contact one another, which is not necessarily the case in experiments (see Fig. 3D in main text). Using this value of , we calculate and the estimated failure strain of the frame as where 0 = 28.7 μm is the pitch of the undeformed helical strand, and s = 34.0 μm is the pitch of the straightened helical strand with all fibers touching. For the case of 29-hr UV-irradiated frames, we can consider UTS − = 0 since they were able to deform beyond Regime 1. Our prediction of the failure strain of the frame , which only considers the straightening of fibers with maximum tensile strain equal to UTS , produces rough estimates of 88.2%, 66.5%, and 59.9% for pristine, 5-hr UV-irradiated, and 29-hr UV-irradiated frames, respectively.
As an alternative to estimate for a knotted frame, we calculated the minimum achievable knot radius of curvature between the braid and the loop for a given fiber UTS , which corresponds to a specific . We define the maximum bending strain of the fiber near the entrance of the knot as follows: where = 6.05 μm is the initial radius of curvature between the braid and the loop at zero frame strain. By replacing with UTS of a pristine, 5-hr UV-irradiated, and 29-hr UVirradiated fibers, we obtained the corresponding : 137% for pristine, 117% for 5-hr UVirradiated, and 102% for 29-hr UV-irradiated frames.
The estimated ~34-47% higher tensile strain at failure for pristine frames compared to 29-hr UVirradiated frames is contrary to our experimental results, where for pristine and UV-irradiated woven frames are close to each other, within 67.4-75.4% (see Fig. 3C). For the knotted frames, despite a trend of increasing with an increasing UTS shown by the samples with the highest for a given UV-irradiation time, most samples still failed within = 74.3-111.4% regardless of their UV-irradiation times. Existing literature on the mechanics of knots have shown that (i) a curvature-based analysis alone is not sufficient to determine the failure properties of tight physical knots without considering localized deformation when contact occurs, (ii) variations in the mechanical properties of the constituent material can change the knot failure mechanism for a given number of fiber crossing points (51)(52)(53), and that (iii) a higher surface friction will result in a higher tensile load vs. strain curve (54). The tensile load vs. strain response of the intertwined frames, informed by the existing literature on overhand knots, serve as the foundation of our prediction of the frame failure mechanism beyond the fiber aligning regime. A further independent study that includes high-fidelity numerical modeling and/or three-dimensional mapping of the fibers during experiments will be needed to predict the frame failure behavior more precisely.
The Degree of Polymer Crosslinking in UV-Irradiated Samples As a guideline to estimate the degree of polymer crosslinking (DC) in our samples, we refer to the work by Bauer et al. (59), which follows a general methodology to determine the DC of a printed structure and analyzes the trends between DC and mechanical properties. While the mechanical characterization of the samples in Ref. (59) mainly focuses on properties obtained from compression testing, a short section in Ref. (59) reports tensile properties of up to 9 pillars printed with varying printing parameters. Assuming the general trends in Ref. (59) hold for our printing parameters, we compare the values of Young's Modulus and ultimate tensile strength (UTS) from our pillar tension tests with those in Ref. (59), and we estimate the DC of our pristine, 5-hr UV-irradiated, and 29-hr UV-irradiated samples to be 39-43%, 43-44%, and 44-45%, respectively.  . (B and C) Tensile responses of a single (B) and a 1x1x2 tessellation (C) of reduced unit cells, where two knotted rhombus frames are assembled in each unit cell with their knot tightening direction aligned with the loading axis. First failure event from each experiment is marked with "x", and relative density of each structure is estimated by comparing the measured fiber radii with * = 1.69 μm for a knotted unit cell with three rhombus frames and a relative density ̅ of 5%. Predicted responses for woven and knotted lattice frames are shown by dashed curves. Scale bars in all SEM images: 20 μm (A), 10 μm (B and C).     Comparison between data correction methods. Load vs. strain data from a rhombus of H = 70 μm tested in tension up to failure and corrected using (i) the compliance correction method and (ii) digital image tracking. The lack of deviation between the two data sets shows that both methods are comparable to one another. Note that charging on polymer surface during prolonged imaging inside an SEM may induce measurement errors when using digital image tracking.

Movie S2.
Video of hierarchical knotted (left, purple) and woven (right, red) rhombuses with designed height H = 70 μm cyclically loaded in tension to an increasing strain value in each subsequent cycle with 300x playback speed. Within the first two cycles, both rhombuses returned to shapes close to their undeformed configurations. In the third cycle, the woven rhombus experienced failure whereas the knotted rhombus was tightened and retained its shape after unloading.

Movie S3.
Video of hierarchical knotted rhombuses of height H = 70 μm (left, dark purple) and 140 μm (right, light purple) loaded in tension up to failure played at 100x speed. The normalized load ( ̅ ) is defined as the applied load divided by the product of the Young's Modulus of pristine IP-Dip and the fiber cross-sectional area.